The problem here is that you have sine instead of [tex]e^{ikx}[/tex] factor.
Use the fact that:
[tex]sin(kx) = \frac{1}{2i}(e^{ikx}-e^{-ikx})[/tex]

Second your answer should be:
[tex]\Psi(x,t) = ... [/tex]
not
[tex]b(k) = ...[/tex]
Look up the source of your "relevant equation" to see what role [tex]b(k)[/tex] plays....also you should give the variable of integration which appears to be k.

Your attempted solution is I suppose integrated over x? The integral you give has a known solution in terms of Dirac delta functions (which relates to my point above)

Remember ultimately you are looking for a solution to the Schrodinger equation which a.) is properly normalized and b.) satisfies the initial condition given.